中点画椭圆算法_中点圆算法

中点画椭圆算法

中点圆算法 (Midpoint circle Algorithm)

This is an algorithm which is used to calculate the entire perimeter points of a circle in a first octant so that the points of the other octant can be taken easily as they are mirror points; this is due to circle property as it is symmetric about its center.

这是一种算法，用于计算第一个八分圆中一个圆的整个周边点，以便可以轻松地将另一个八分圆的点视为镜像点；这是由于圆的属性有关它的中心对称。

In this algorithm decision parameter is based on a circle equation. As we know that the equation of a circle is x² +y² =r² when the centre is (0, 0).

在该算法中，决策参数基于圆方程。众所周知，当中心为(0，0)时，圆的方程为x ² + y ² = r ² 。

Now let us define the function of a circle i.e.: fcircle(x,y)= x² +y² - r²

现在让我们定义一个圆的函数，即： fcircle(x，y)= x ² + y ² -r ²

If fcircle < 0 then x, y is inside the circle boundary.

如果fcircle <0，则x ， y在圆边界之内。
If fcircle > 0 then x, y is outside the circle boundary.

如果fcircle> 0，则x ， y在圆边界之外。
If fcircle = 0 then x, y is on the circle boundary.

如果fcircle = 0，则x ， y在圆边界上。

决策参数 (Decision parameter)

p_k =fcircle(x_k+1,y_k-1/2) where p_k is a decision parameter and in this ½ is taken because it is a midpoint value through which it is easy to calculate value of y_k and y_k-1.

p _k = fcircle(x _{k + 1} ，y _{k-1 / 2} ) ，其中p _k是决策参数，在此1/2中采用p _k是因为它是一个中点值，通过该中点值很容易计算y _k和y _{k -1} 。

I.e. p_k= (x_k+1)²+ (y_k-1/2)²-r²

即p _k =(x _{k + 1} ) ² +(y _{k-1 / 2} ) ² -r ²

If p_k <0 then midpoint is inside the circle in this condition we select y is y_k otherwise we will select next y as y_k-1 for the condition of p_k > 0.

如果p _k <0，则在这种情况下中点在圆内，我们选择y为y _k，否则对于p _k > 0的情况，我们将下一个y选择为y _k-1 。

结论 (Conclusion)

If p_k < 0 then y_k+1=y_k, by this the plotting points will be ( x_k+1 ,y_k). By this the value for the next point will be given as:

如果p _k <0，则y _{k + 1} = y _k ，由此绘制点将为(x _{k + 1} ，y _k ) 。这样，下一点的值将为：

P_k+1=p_k +2(x_k+1) +1

P _{k + 1} = p _k +2(x _{k + 1} )+1
If p_k > 0 then y_k+1=y_k-1, by this the plotting points will be (x_k+1, y_k-1). By this the value of the next point will be given as:

如果p _k > 0，则y _{k + 1} = y _k-1 ，由此绘制点将为(x _{k + 1} ，y _k-1 ) 。这样，下一点的值将为：

P_k+1=p_k+2(x_k+1) +1-2(y_k+1)

P _{k + 1} = p _k +2(x _{k + 1} )+ 1-2(y _{k + 1} )

初始决策参数 (Initial decision parameter)

P₀ = fcircle (1, r-1/2)

P ₀ =圆(1，r-1 / 2)

This is taken because of (x₀, y₀) = (0, r)

这是因为(x ₀ ，y ₀ )=(0，r)

i.e. p₀ =5/4-r or 1-r, (1-r will be taken if r is integer)

即p ₀ = 5 / 4-r或1-r ，(如果r为整数则采用1-r )

算法 (ALGORITHM)

In this the input radius r is there with a centre (x_c , y_c). To obtain the first point m the circumference of a circle is centered on the origin as (x₀,y₀) = (0,r).

在此，输入半径r以一个中心(x _c ，y _c )为中心。为了获得第一个点m ，圆的圆周以(x ₀ ，y ₀ )=(0，r)为中心 。
Calculate the initial decision parameters which are:

计算初始决策参数为：

p₀ =5/4-r or 1-r

p ₀ = 5 / 4-r或1-r
Now at each x_k position starting k=0, perform the following task.

现在，在从k = 0开始的每个x _k位置，执行以下任务。

if

如果

p_k < 0 then plotting point will be ( x_k+1 ,y_k) and

p _k <0，则绘图点将为(x _{k + 1} ，y _k )并且

P_k+1=p_k +2(x_k+1) +1

P _{k + 1} = p _k +2(x _{k + 1} )+1

else the next point along the circle is (x

否则沿圆的下一个点是[x

_k+1, y_k-1) and

_{k + 1} ，y _k-1 )和

P_k+1=p_k+2(x_k+1) +1-2(y_k+1)

P _{k + 1} = p _k +2(x _{k + 1} )+ 1-2(y _{k + 1} )
Determine the symmetry points in the other quadrants.

确定其他象限中的对称点。
Now move at each point by the given centre that is:

现在按照给定的中心在每个点处移动：

x=x+x_c

x = x + x _c

y=y+y_c

y = y + y _c
At last repeat steps from 3 to 5 until the condition x>=y.

最后重复步骤3到5，直到条件x> = y为止。

翻译自: https://www.includehelp.com/algorithms/midpoint-circle.aspx

中点画椭圆算法